Linear Nonhomogeneous Systems of Differential Equations with Constant Coefficients
A normal linear inhomogeneous system of n equations with constant coefficients can be written as
where t is the independent variable (often t is time), xi (t) are unknown functions which are continuous and differentiable on an interval [a, b] of the real number axis t, aij (i, j = 1, ..., n) are the constant coefficients, fi (t) are given functions of the independent variable t. We assume that the functions xi (t), fi (t) and the coefficients aij may take both real and complex values.
We introduce the following vectors:
and the square matrix
Then the system of equations can be written in a more compact matrix form as
For nonhomogeneous linear systems, as well as in the case of a linear homogeneous equation, the following important theorem is valid:
The general solution
Methods of solutions of the homogeneous systems are considered on other web-pages of this section. Therefore, below we focus primarily on how to find a particular solution.
Another important property of linear inhomogeneous systems is the principle of superposition, which is formulated as follows:
If
is a solution of the system with the inhomogeneous part
The most common methods of solution of the nonhomogeneous systems are the method of elimination, the method of undetermined coefficients (in the case where the function
Elimination Method
This method allows to reduce the normal nonhomogeneous system of
Method of Undetermined Coefficients
The method of undetermined coefficients is well suited for solving systems of equations, the inhomogeneous part of which is a quasi-polynomial.
A real vector quasi-polynomial is a vector function of the form
where
where
In the case when the inhomogeneous part
For example, if the nonhomogeneous function is
a particular solution should be sought in the form
where
Similar rules for determining the degree of the polynomials are used for quasi-polynomials of kind
Here the resonance case occurs when the number
After the structure of a particular solution
Method of Variation of Constants
The method of variation of constants (Lagrange method) is the common method of solution in the case of an arbitrary right-hand side
Suppose that the general solution of the associated homogeneous system is found and represented as
where
We replace the constants
Since the Wronskian of the system is not equal to zero, then there exists the inverse matrix
where
Then the general solution of the nonhomogeneous system can be written as
We see that a particular solution of the nonhomogeneous equation is represented by the formula
Thus, the solution of the nonhomogeneous equation can be expressed in quadratures for any inhomogeneous term